Problem 43 - Algebra for College Students

Question

Bill, Carol, and Alice are to be seated at random in a row of three seats. Find the probability that Bill and Carol will be seated side by side. \(\frac{2}{3}\)

Step-by-Step Solution

Answer

The probability is \(\frac{2}{3}\).

Step 1: Determine Total Possible Arrangements

First, calculate the total number of ways to arrange Bill, Carol, and Alice in a row of three seats. Since we have 3 people, the total number of arrangements is given by the factorial of 3, which is:\[3! = 3 \times 2 \times 1 = 6\]So, there are 6 possible ways to seat Bill, Carol, and Alice.

Step 2: Group Bill and Carol Together

Next, treat Bill and Carol as a single unit or 'block' to ensure they sit together. This block can be seated in any of the 2 positions (leftmost or middle positions) as the pair swaps positions in each spot. The possible arrangements then involve 2 items: the block and Alice.Within the block, Bill and Carol can swap seats, which allows their internal arrangements. Therefore, the combinations are:\[2! \times 2! = 2 \times 2 = 4\]This is calculated by 2 positions for the block and 2 swap arrangements inside the block.

Step 3: Calculate the Probability

To find the probability that Bill and Carol sit together, divide the number of successful arrangements (where Bill and Carol sit together) by the total number of arrangements:\[\text{Probability} = \frac{\text{Number of successful arrangements}}{\text{Total arrangements}} = \frac{4}{6} = \frac{2}{3}\]

Step 4: Conclusion

Thus, the probability that Bill and Carol will be seated side by side is \(\frac{2}{3}\).

Key Concepts

Combinatorics

Combinatorics is a fascinating branch of mathematics that deals with counting, arranging, and grouping elements within a set. It's like solving a puzzle to find how many different ways you can organize or choose items. This field is crucial for solving probability problems, as it allows us to calculate possible outcomes and desired events.
One of the core ideas in combinatorics is determining the total number of arrangements (or combinations) for a given set of elements. This is particularly useful when you need to figure out how many ways you can sequence or select items from a set.

• If you have a set of distinct objects, you often use permutations to find the number of possible arrangements.
• If you want to count how many ways you can choose a subset of a set without regard to the order, you use combinations.

In our exercise case, combinatorics helps us find the total arrangements of Bill, Carol, and Alice to eventually determine the probability of them sitting side-by-side. Understanding combinatorics not only enhances your problem-solving skills but also helps in developing logical reasoning, which is applicable in various real-world situations.

Factorial Concept

The factorial concept is a fundamental tool in combinatorics. It's represented by an exclamation mark after a number, like this: \(n!\). Factorials help calculate permutations and combinations by providing the total number of ways to arrange a set of \(n\) distinct objects.
Here's how it works:

• For any positive integer \(n\), the factorial \(n!\) is defined as the product of all positive integers less than or equal to \(n\).
• For example, \(3! = 3 \times 2 \times 1 = 6\), which means there are 6 ways to arrange three distinct people in three seats.
• The factorial of 0 is defined to be 1, i.e., \(0! = 1\).

In the original exercise, using the factorial concept allowed us to quickly calculate the total number of seating permutations for Bill, Carol, and Alice. Grasping this concept ensures that you're well-equipped to handle complex counting problems both in probability and other areas of mathematics.

Permutations

Permutations refer to the different ways of arranging all items from a given set where the order matters. Think of a permutation as a specific sequence or order in which distinct objects can be arranged. This concept is particularly important in scenarios where positioning is crucial.
Understanding permutations involves a few key points:

• Permutations of a set can be calculated using factorial notation, where \(n!\) gives the total arrangements of \(n\) items.
• When only a part of the set is used, the formula changes to \(\frac{n!}{(n-r)!}\), where \(r\) is the number of positions being used.
• For example, if you need to arrange two people out of three in seats, you'd consider \(\frac{3!}{(3-2)!} = 6\) permutations.

In our exercise, permutations were used to find the number of ways Bill and Carol could be placed next to each other, treating them as a single block, which revealed the arrangements that satisfied our condition. Mastering permutations enables you to tackle similar probability problems where the sequence significantly impacts the outcome.